WSWF translations (TODO explicit form of the ops)

Former-commit-id: 44f286281af3f44208c2b1e2aabbe7234b0cd675
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Marek Nečada 2019-07-20 13:54:17 +03:00
parent b259bd46b4
commit e26ca79a53
2 changed files with 595 additions and 1 deletions

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\end_inset \end_inset
\begin_inset FormulaMacro
\newcommand{\uvec}[1]{\mathbf{\hat{#1}}}
\end_inset
\begin_inset FormulaMacro \begin_inset FormulaMacro
\newcommand{\ud}{\mathrm{d}} \newcommand{\ud}{\mathrm{d}}
\end_inset \end_inset
@ -162,6 +167,16 @@
\end_inset \end_inset
\begin_inset FormulaMacro
\newcommand{\vsh}[3]{\vect A_{#1,#2,#3}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\vshD}[3]{\vect A'_{#1,#2,#3}}
\end_inset
\begin_inset FormulaMacro \begin_inset FormulaMacro
\newcommand{\hgfr}{\mathbf{F}} \newcommand{\hgfr}{\mathbf{F}}
\end_inset \end_inset
@ -213,7 +228,7 @@
\begin_inset FormulaMacro \begin_inset FormulaMacro
\newcommand{\particle}{\mathrm{\Omega}} \newcommand{\particle}{\mathrm{\Theta}}
\end_inset \end_inset
@ -232,6 +247,71 @@
\end_inset \end_inset
\begin_inset FormulaMacro
\newcommand{\rcoeffp}[1]{a_{#1}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\rcoeffptlm}[4]{\rcoeffp{#1}_{#2#3#4}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\vswfrtlm}[3]{\vect v_{#1#2#3}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\outcoeffp}[1]{f_{#1}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\outcoeffptlm}[4]{\outcoeffp{#1}_{#2#3#4}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\vswfouttlm}[3]{\vect u_{#1#2#3}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\Tp}[1]{T_{#1}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\openball}[2]{B_{#1}\left(#2\right)}
\end_inset
\begin_inset FormulaMacro
\newcommand{\closedball}[2]{B_{#1}#2}
\end_inset
\begin_inset FormulaMacro
\newcommand{\tropr}{\mathcal{R}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\troprp}[2]{\mathcal{\tropr}_{#1\leftarrow#2}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\trops}{\mathcal{S}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\tropsp}[2]{\mathcal{\trops}_{#1\leftarrow#2}}
\end_inset
\end_layout \end_layout
\begin_layout Title \begin_layout Title
@ -450,6 +530,16 @@ filename "finite.lyx"
\end_inset \end_inset
\end_layout
\begin_layout Standard
\begin_inset CommandInset include
LatexCommand include
filename "finite-cs.lyx"
\end_inset
\end_layout \end_layout
\begin_layout Standard \begin_layout Standard

504
lepaper/finite-cs.lyx Normal file
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#LyX 2.1 created this file. For more info see http://www.lyx.org/
\lyxformat 474
\begin_document
\begin_header
\textclass article
\use_default_options true
\maintain_unincluded_children false
\language finnish
\language_package default
\inputencoding auto
\fontencoding global
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\font_sf_scale 100
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\graphics default
\default_output_format pdf4
\output_sync 0
\bibtex_command default
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\spacing single
\use_hyperref true
\pdf_title "Sähköpajan päiväkirja"
\pdf_author "Marek Nečada"
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\use_package amsmath 1
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\use_package mathdots 1
\use_package mathtools 1
\use_package mhchem 1
\use_package stackrel 1
\use_package stmaryrd 1
\use_package undertilde 1
\cite_engine basic
\cite_engine_type default
\biblio_style plain
\use_bibtopic false
\use_indices false
\paperorientation portrait
\suppress_date false
\justification true
\use_refstyle 1
\index Index
\shortcut idx
\color #008000
\end_index
\secnumdepth 3
\tocdepth 3
\paragraph_separation indent
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\quotes_language swedish
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\end_header
\begin_body
\begin_layout Subsection
Dual vector spherical harmonics
\end_layout
\begin_layout Standard
For evaluation of expansion coefficients of incident fields, it is useful
to introduce „dual“ vector spherical harmonics
\begin_inset Formula $\vshD{\tau}lm$
\end_inset
defined by duality relation
\begin_inset Formula
\begin{equation}
\iint\vshD{\tau'}{l'}{m'}\left(\uvec r\right)\cdot\vsh{\tau}lm\left(\uvec r\right)\,\ud\Omega=\delta_{\tau'\tau}\delta_{l'l}\delta_{m'm}\label{eq:dual vsh}
\end{equation}
\end_inset
(complex conjugation not implied in the dot product here).
In our convention, we have
\begin_inset Formula
\[
\vshD{\tau}lm\left(\uvec r\right)=\left(\vsh{\tau}lm\left(\uvec r\right)\right)^{*}=\left(-1\right)^{m}\vsh{\tau}{l-}m\left(\uvec r\right).
\]
\end_inset
\end_layout
\begin_layout Subsection
Translation operators
\end_layout
\begin_layout Standard
Let
\begin_inset Formula $\vect r_{1},\vect r_{2}$
\end_inset
be two different origins; a regular VSWF with origin
\begin_inset Formula $\vect r_{1}$
\end_inset
can be always expanded in terms of regular VSWFs with origin
\begin_inset Formula $\vect r_{2}$
\end_inset
as follows:
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
\vswfrtlm{\tau}lm\left(k\left(\vect r-\vect r_{1}\right)\right)=\sum_{\tau'l'm'}\tropr_{\tau lm;\tau'l'm'}\left(k\left(\vect r_{2}-\vect r_{1}\right)\right)\vswfrtlm{\tau'}{l'}{m'}\left(\vect r-\vect r_{2}\right),\label{eq:regular vswf translation}
\end{equation}
\end_inset
where an explicit formula for the (regular)
\emph on
translation operator
\emph default
\begin_inset Formula $\tropr$
\end_inset
reads in eq.
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:translation operator"
\end_inset
below.
For singular (outgoing) waves, the form of the expansion differs inside
and outside the ball
\begin_inset Formula $\openball{\left\Vert \vect r_{2}-\vect r_{1}\right\Vert }{\vect r_{1}}:$
\end_inset
\begin_inset Formula
\begin{eqnarray}
\vswfouttlm{\tau}lm\left(k\left(\vect r-\vect r_{1}\right)\right) & = & \begin{cases}
\sum_{\tau'l'm'}\trops_{\tau lm;\tau'l'm'}\left(k\left(\vect r_{2}-\vect r_{1}\right)\right)\vswfouttlm{\tau'}{l'}{m'}\left(\vect r-\vect r_{2}\right), & \vect r\in\openball{\left\Vert \vect r_{2}-\vect r_{1}\right\Vert }{\vect r_{1}}\\
\sum_{\tau'l'm'}\tropr_{\tau lm;\tau'l'm'}\left(k\left(\vect r_{2}-\vect r_{1}\right)\right)\vswfrtlm{\tau'}{l'}{m'}\left(\vect r-\vect r_{2}\right), & \vect r\notin\closedball{\left\Vert \vect r_{2}-\vect r_{1}\right\Vert }{\left|\vect r_{1}\right|}
\end{cases},\label{eq:singular vswf translation}
\end{eqnarray}
\end_inset
where the singular translation operator
\begin_inset Formula $\trops$
\end_inset
has the same form as
\begin_inset Formula $\tropr$
\end_inset
in
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:translation operator"
\end_inset
except the regular spherical Bessel functions
\begin_inset Formula $j_{l}$
\end_inset
are replaced with spherical Hankel functions
\begin_inset Formula $h_{l}^{(1)}$
\end_inset
.
\begin_inset Note Note
status open
\begin_layout Plain Layout
TODO note about expansion exactly on the sphere.
\end_layout
\end_inset
\end_layout
\begin_layout Standard
As MSTMM deals most of the time with the
\emph on
expansion coefficients
\emph default
of fields
\begin_inset Formula $\rcoeffptlm p{\tau}lm,\outcoeffptlm p{\tau}lm$
\end_inset
in different origins
\begin_inset Formula $\vect r_{p}$
\end_inset
rather than with the VSWFs directly, let us write down how
\emph on
they
\emph default
transform under translation.
Let us assume the field can be in terms of regular waves everywhere, and
expand it in two different origins
\begin_inset Formula $\vect r_{p},\vect r_{q}$
\end_inset
,
\begin_inset Formula
\[
\vect E\left(\vect r,\omega\right)=\sum_{\tau,l,m}\rcoeffptlm p{\tau}lm\vswfrtlm{\tau}lm\left(k\left(\vect r-\vect r_{p}\right)\right)=\sum_{\tau',l',m'}\rcoeffptlm q{\tau'}{l'}{m'}\vswfrtlm{\tau}{'l'}{m'}\left(k\left(\vect r-\vect r_{q}\right)\right).
\]
\end_inset
Re-expanding the waves around
\begin_inset Formula $\vect r_{p}$
\end_inset
in terms of waves around
\begin_inset Formula $\vect r_{q}$
\end_inset
using
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:regular vswf translation"
\end_inset
,
\begin_inset Formula
\[
\vect E\left(\vect r,\omega\right)=\sum_{\tau,l,m}\rcoeffptlm p{\tau}lm\sum_{\tau'l'm'}\tropr_{\tau lm;\tau'l'm'}\left(k\left(\vect r_{q}-\vect r_{p}\right)\right)\vswfrtlm{\tau'}{l'}{m'}\left(\vect r-\vect r_{q}\right)
\]
\end_inset
and comparing to the original expansion around
\begin_inset Formula $\vect r_{q}$
\end_inset
, we obtain
\begin_inset Formula
\begin{equation}
\rcoeffptlm q{\tau'}{l'}{m'}=\sum_{\tau,l,m}\tropr_{\tau lm;\tau'l'm'}\left(k\left(\vect r_{q}-\vect r_{p}\right)\right)\rcoeffptlm p{\tau}lm.\label{eq:regular vswf coefficient translation}
\end{equation}
\end_inset
For the sake of readability, we introduce a shorthand matrix form for
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:regular vswf coefficient translation"
\end_inset
\begin_inset Formula
\begin{equation}
\rcoeffp q=\troprp qp\rcoeffp p\label{eq:reqular vswf coefficient vector translation}
\end{equation}
\end_inset
(note the reversed indices; TODO redefine them in
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:regular vswf translation"
\end_inset
,
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:singular vswf translation"
\end_inset
? Similarly, if we had only outgoing waves in the original expansion around
\begin_inset Formula $\vect r_{p}$
\end_inset
, we would get
\begin_inset Formula
\begin{equation}
\rcoeffp q=\tropsp qp\outcoeffp p\label{eq:singular to regular vswf coefficient vector translation}
\end{equation}
\end_inset
for the expansion inside the ball
\begin_inset Formula $\openball{\left\Vert \vect r_{q}-\vect r_{p}\right\Vert }{\vect r_{p}}$
\end_inset
\begin_inset Note Note
status open
\begin_layout Plain Layout
CHECKME
\end_layout
\end_inset
and
\begin_inset Formula
\begin{equation}
\outcoeffp q=\troprp qp\outcoeffp p\label{eq:singular to singular vswf coefficient vector translation-1}
\end{equation}
\end_inset
outside.
\end_layout
\begin_layout Standard
The translation operator can be expressed explicitly as
\begin_inset Formula
\begin{equation}
\tropr_{\tau lm;\tau'l'm'}\left(\vect d\right)=\dots.\label{eq:translation operator}
\end{equation}
\end_inset
\end_layout
\begin_layout Subsection
Plane wave expansion coefficients
\end_layout
\begin_layout Standard
A transversal (
\begin_inset Formula $\vect k\cdot\vect E_{0}=0$
\end_inset
) plane wave propagating in direction
\begin_inset Formula $\uvec k$
\end_inset
with (complex) amplitude
\begin_inset Formula $\vect E_{0}$
\end_inset
can be expanded into regular VSWFs [REF KRIS]
\begin_inset Formula
\[
\vect E_{\mathrm{PW}}\left(\vect r,\omega\right)=\vect E_{0}e^{ik\uvec k\cdot\vect r}=\sum_{\tau,l,m}\rcoeffptlm{}{\tau}lm\left(\vect k,\vect E_{0}\right)\vswfrtlm{\tau}lm\left(k\vect r\right),
\]
\end_inset
with expansion coefficients
\begin_inset Formula
\begin{eqnarray}
\rcoeffptlm{}1lm\left(\vect k,\vect E_{0}\right) & = & 4\pi i^{l}\vshD 1lm\left(\uvec k\right),\nonumber \\
\rcoeffptlm{}2lm\left(\vect k,\vect E_{0}\right) & = & -4\pi i^{l+1}\vshD 2lm\left(\uvec k\right).\label{eq:plane wave expansion}
\end{eqnarray}
\end_inset
\end_layout
\begin_layout Subsection
Cross-sections (single-particle)
\end_layout
\begin_layout Standard
Extinction, scattering and absorption cross sections of a single particle
irradiated by a plane wave propagating in direction
\begin_inset Formula $\uvec k$
\end_inset
are
\begin_inset CommandInset citation
LatexCommand cite
after "sect. 7.8.2"
key "kristensson_scattering_2016"
\end_inset
\begin_inset Formula
\begin{eqnarray}
\sigma_{\mathrm{ext}}\left(\uvec k\right) & = & -\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\Re\left(\rcoeffp{}^{\dagger}\outcoeffp{}\right)=-\frac{1}{2k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\rcoeffp{}^{\dagger}\left(\Tp{}+\Tp{}^{\dagger}\right)\rcoeffp{},\label{eq:extincion CS single}\\
\sigma_{\mathrm{scat}}\left(\uvec k\right) & = & \frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\left\Vert \outcoeffp{}\right\Vert ^{2}=\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\rcoeffp{}^{\dagger}\left(\Tp{}^{\dagger}\Tp{}\right)\rcoeffp{},\label{eq:scattering CS single}\\
\sigma_{\mathrm{abs}}\left(\uvec k\right) & = & \sigma_{\mathrm{ext}}\left(\uvec k\right)-\sigma_{\mathrm{scat}}\left(\uvec k\right)=-\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\left(\Re\left(\rcoeffp{}^{\dagger}\outcoeffp{}\right)+\left\Vert \outcoeffp{}\right\Vert ^{2}\right)\nonumber \\
& & =\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\rcoeffp{}^{\dagger}\left(\Tp{}^{\dagger}\Tp{}+\frac{\Tp{}^{\dagger}+\Tp{}}{2}\right)\rcoeffp{},\label{eq:absorption CS single}
\end{eqnarray}
\end_inset
where
\begin_inset Formula $\rcoeffp{}=\rcoeffp{}\left(\vect k,\vect E_{0}\right)$
\end_inset
is the vector of plane wave expansion coefficients as in
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:plane wave expansion"
\end_inset
.
\end_layout
\begin_layout Standard
For a system of many scatterers, Kristensson derives only the scattering
cross section formula
\begin_inset Formula
\[
\sigma_{\mathrm{scat}}\left(\uvec k\right)=\frac{1}{k^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\sum_{p\in\mathcal{P}}\left\Vert \outcoeffp p\right\Vert ^{2}.
\]
\end_inset
Let us derive the many-particle scattering and absorption cross sections.
First, let us take a ball circumscribing all the scatterers at once,
\begin_inset Formula $\openball R{\vect r_{\square}}\supset\particle$
\end_inset
.
Outside
\begin_inset Formula $\openball R{\vect r_{\square}}$
\end_inset
, we can describe the EM fields as if there was only a single scatterer,
\begin_inset Formula
\[
\vect E\left(\vect r\right)=\sum_{\tau,l,m}\left(\rcoeffptlm{\square}{\tau}lm\vswfrtlm{\tau}lm\left(\vect r-\vect r_{\square}\right)+\outcoeffptlm{\square}{\tau}lm\vswfouttlm{\tau}lm\left(\vect r-\vect r_{\square}\right)\right),
\]
\end_inset
where
\begin_inset Formula $\rcoeffp{\square},\outcoeffp{\square}$
\end_inset
are the vectors of VSWF expansion coefficients of the incident and total
scattered fields, respectively, at origin
\begin_inset Formula $\vect r_{\square}$
\end_inset
.
In principle, one could evaluate
\begin_inset Formula $\outcoeffp{\square}$
\end_inset
using the translation operators (REF!!!) and use the single-scatterer formulae
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:extincion CS single"
\end_inset
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:absorption CS single"
\end_inset
to obtain the cross sections.
However, this is not suitable for numerical evaluation with truncation
in multipole degree; hence we need to express them in terms of particle-wise
expansions
\begin_inset Formula $\rcoeffp p,\outcoeffp p$
\end_inset
.
\end_layout
\end_body
\end_document